Question
Features of Graphs of Quadratic Equations Quick Check
4 of 54 of 5 Items
Question
Create a graph; then answer the following question.
What are the key points on the graph of y=x2−8x? Name the vertex, x-intercept(s), and y-intercept.
(1 point)
Responses
x-intercept: (4,−16)
y-intercept: (0,0)
vertex: (8,0)
x -intercept: left parenthesis 4 comma negative 16 right parenthesis y -intercept: left parenthesis 0 comma 0 right parenthesis vertex: left parenthesis 8 comma 0 right parenthesis
x-intercepts: (0,0) (8,0)
y-intercept: (0,0)
vertex: (4,−16)
x -intercepts: left parenthesis 0 comma 0 right parenthesis left parenthesis 8 comma 0 right parenthesis y -intercept: left parenthesis 0 comma 0 right parenthesis vertex: left parenthesis 4 comma negative 16 right parenthesis
x-intercepts: (0,0) (0,8)
vertex: (4,−16)
There is no y-intercept.
x -intercepts: left parenthesis 0 comma 0 right parenthesis left parenthesis 0 comma 8 right parenthesis vertex: left parenthesis 4 comma negative 16 right parenthesis There is no y -intercept.
x-intercept: (0,8)
y-intercept: (0,0)
vertex: (4,−16)
x -intercept: left parenthesis 0 comma 8 right parenthesis y -intercept: left parenthesis 0 comma 0 right parenthesis vertex: left parenthesis 4 comma negative 16 right parenthesis
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Answers
GPT-4o mini
To analyze the quadratic equation \( y = x^2 - 8x \), we can determine its key features such as the vertex, x-intercepts, and y-intercept.
1. **Vertex**:
To find the vertex of the quadratic equation, we can use the vertex formula \( x = -\frac{b}{2a} \) where \( a = 1 \) and \( b = -8 \).
\[
x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4
\]
Now, we substitute \( x = 4 \) back into the equation to find \( y \):
\[
y = (4)^2 - 8(4) = 16 - 32 = -16
\]
So, the vertex is \( (4, -16) \).
2. **X-intercepts**:
To find the x-intercepts, we set \( y = 0 \):
\[
0 = x^2 - 8x
\]
Factoring gives us:
\[
0 = x(x - 8)
\]
This gives us two x-intercepts:
\[
x = 0 \quad \text{and} \quad x = 8
\]
Therefore, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).
3. **Y-intercept**:
To find the y-intercept, we set \( x = 0 \):
\[
y = (0)^2 - 8(0) = 0
\]
So, the y-intercept is \( (0, 0) \).
### Summary of Key Points
- Vertex: \( (4, -16) \)
- X-intercepts: \( (0, 0) \) and \( (8, 0) \)
- Y-intercept: \( (0, 0) \)
### Responses to Choose From
The correct features you can choose from the options provided are:
- x-intercepts: \( (0, 0) \) and \( (8, 0) \)
- y-intercept: \( (0, 0) \)
- vertex: \( (4, -16) \)
Make sure to select the option that accurately represents this information!
1. **Vertex**:
To find the vertex of the quadratic equation, we can use the vertex formula \( x = -\frac{b}{2a} \) where \( a = 1 \) and \( b = -8 \).
\[
x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4
\]
Now, we substitute \( x = 4 \) back into the equation to find \( y \):
\[
y = (4)^2 - 8(4) = 16 - 32 = -16
\]
So, the vertex is \( (4, -16) \).
2. **X-intercepts**:
To find the x-intercepts, we set \( y = 0 \):
\[
0 = x^2 - 8x
\]
Factoring gives us:
\[
0 = x(x - 8)
\]
This gives us two x-intercepts:
\[
x = 0 \quad \text{and} \quad x = 8
\]
Therefore, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).
3. **Y-intercept**:
To find the y-intercept, we set \( x = 0 \):
\[
y = (0)^2 - 8(0) = 0
\]
So, the y-intercept is \( (0, 0) \).
### Summary of Key Points
- Vertex: \( (4, -16) \)
- X-intercepts: \( (0, 0) \) and \( (8, 0) \)
- Y-intercept: \( (0, 0) \)
### Responses to Choose From
The correct features you can choose from the options provided are:
- x-intercepts: \( (0, 0) \) and \( (8, 0) \)
- y-intercept: \( (0, 0) \)
- vertex: \( (4, -16) \)
Make sure to select the option that accurately represents this information!